Abstract

We consider random interlacements on ${\mathbb{Z}}^d$, $d\ge 3$, when their vacant set is in a strongly percolative regime. Given a large box centered at the origin, we establish an asymptotic upper bound on the exponential rate of decay of the probability that the box contains an excessive fraction $\nu$ of points that are disconnected by random interlacements from the boundary of a concentric box of double size. As an application, we show that when $\nu$ is not too large this asymptotic upper bound matches the asymptotic lower bound derived by Sznitman [arXiv:1906.05809], and the exponential rate of decay is governed by the variational problem in the continuum involving the percolation function of the vacant set of random interlacements that he studied [arXiv:1910.04737]. This is a further confirmation of the pertinence of this variational problem.

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